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The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. (English) Zbl 1130.35061
Summary: We consider the existence and non-existence of non-trivial solutions to quasilinear Brezis-Nirenberg-type problems with singular weights. First, we shall obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pokhozhaev-type identity and obtain a non-existence result. Thirdly, thanks to the generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS) c condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.
35J60Nonlinear elliptic equations